Yours is a figurative example that draws a comparison to supposedly dangerous animals. To say that a person does not or will not bite means there is no cause for fear, anxiety, timidity or discomfort concerning that person. The person is benign.
Although it does roughly mean “to have a harmful effect,” I think it deserves its own consideration as a use because it is a fixed saying unto itself.
Bite normally means to close your teeth on something (to bite a hamburger).
“I won’t bite” is a friendly way of saying “I’m not going to be be cruel, nasty or horrible. I’m going to be open-minded, so don’t worry.”
It means “Don’t be afraid of me”. Could be said by a potential lover!
“I won’t scare you” is not quite right. “I won’t hurt you” is more appropriate, but it is never used in a serious way. There is never any REAL danger.
Tofu, it can be used in a situation where real fear exists, but it is most commonly used where someone is timid, aloof or nervous. As the last poster has said, there is never any real danger. It is a facetious way to highlight the absurdity or strangeness of behaving as if there were.
Hi, Sarah. I haven’t seen you in years, and you’ve been over there all night. Sit by me. I won’t bite.
It might be said to a toddler or child who is afraid of a specific person.
Ally, go over to your Uncle John. Stop crying. He won’t bite.